Minimal Solutions For A Class Of Elliptic Systems

There exists a set $\cal U$ in the plane, such that elements of $\cal U$ correspond to minimal stable solutions of a two-parameter elliptic system. For points on the boundary $\ Upsilon$ of $\cal U$, there exist weak solutions to the elliptic system. The paper studies the properties of the curve $\Upsilon$. © 2005 London Mathematical Society.373405416Brézis, H., Cabré, X., Some simple nonlinear PDE's without solutions (1998) Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 1, pp. 223-262Brézis, H., Cazenave, T., Martel, Y., Ramiandrisoa, A., Blow up for ut - Δu = g(u) revisited (1996) Adv. Differential Equations, 1, pp. 73-90Cantrell, R.S., Cosner, C., On the positone problem for elliptic systems (1985) Indiana Univ. Math. J., 34, pp. 517-532Cantrell, R.S., Cosner, C., On the convex case in the positone problem for elliptic systems (1988) Nonlinear Anal., 12, pp. 827-853Cohen, D.S., Keller, H.B., Some positone problems suggested by nonlinear heat generation (1967) J. Math. Mech., 16, pp. 1361-1376Crandall, M.G., Rabinowitz, P.H., Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems (1975) Arch. Ration. Mech. Anal., 58, pp. 207-218Dávila, J., Montenegro, M., Positive versus free boundary solutions to a singular equation (2003) J. Anal. Math., 90, pp. 303-335Fujita, H., On the nonlinear equations Δu + eu = 0 and ∂u/∂t = Δu + eu (1969) Bull. Amer. Math. Soc., 75, pp. 132-135Hess, P., On the eigenvalue problem for weakly coupled elliptic systems (1983) Arch. Rational Mech. Anal., 81, pp. 151-159Hudjaev, S.I., Boundary value problems for certain quasilinear elliptic equations (1964) Dokl. Akad. Nauk SSSR, 154, pp. 787-790Kato, T., (1976) Perturbation Theory for Linear Operators, 2nd Edn, , Grund. Math. Wiss. 132 (Springer, Berlin/New York)Krasnoselskii, M.A., (1964) Positive Solutions of Operator Equations, , P. Noordhoff Ltd., GroningenLeung, A., (1989) Systems of Nonlinear Partial Differential Equations. Applications to Biology and Engineering, , Math. Appl. (Kluwer Academic Publishers, Dordrecht)López-Gómez, J., Molina-Meyer, M., The maximum principle for cooperative weakly coupled elliptic systems and some applications (1994) Differential Integral Equations, 7, pp. 383-398Montenegro, M., The construction of principal spectral curves for Lane-Emden systems and applications (2000) Ann. Scuola Norm. Sup. Pisa Cl. Sci., 29, pp. 193-229Sweers, G., Strong positivity in C(Ω) for elliptic systems (1992) Math. Z., 209, pp. 251-27